The prime symbol (′) is a common mathematical notation used to denote a variable, such as `a′` or `v′`, that is derived from or related to another variable (`a` or `v`) without needing a new name. This notation, read as 'a-prime' or 'v-prime', can signify a value after a transformation, a derivative, or a subsequent state. It is frequently combined with subscripts, as in `v_i'`, which represents the prime version of the i-th element `v_i`.

Prime Notation in Mathematics

In mathematics and deep learning, vectors are typically distinguished from scalars through specific typographical conventions. A standard approach is to denote a vector using a bold lowercase letter, such as $$\mathbf{x}$$, $$\mathbf{y}$$, or $$\mathbf{z}$$, which visually differentiates it from an unbolded scalar variable like $$x$$.

Typographical Conventions for Vectors

|$$\KaTeX$$ |`Plain Text`| | ------- | ------- | |$$\Alpha$$ | `\Alpha`| |$$\Beta$$ | `\Beta`| |$$\Gamma$$ | `\Gamma`| |$$\Delta$$ | `\Delta`| |$$\Epsilon$$ | `\Epsilon`| |$$\Zeta$$ | `\Zeta`| |$$\Eta$$ | `\Eta`| |$$\Theta$$ | `\Theta`| |$$\Iota$$ | `\Iota`| |$$\Kappa$$ | `\Kappa`| |$$\Lambda$$ | `\Lambda`| |$$\Mu$$ | `\Mu`| |$$\Nu$$ | `\Nu`| |$$\Xi$$ | `\Xi`| |$$\Omicron$$ | `\Omicron`| |$$\Pi$$ | `\Pi`| |$$\Rho$$ | `\Rho`| |$$\Sigma$$ | `\Sigma`| |$$\Tau$$ | `\Tau`| |$$\Upsilon$$ | `\Upsilon`| |$$\Phi$$ | `\Phi`| |$$\Chi$$ | `\Chi`| |$$\Psi$$ | `\Psi`| |$$\Omega$$ | `\Omega`| |$$\varGamma$$ | `\varGamma`| |$$\varDelta$$ | `\varDelta`| |$$\varTheta$$ | `\varTheta`| |$$\varLambda$$ | `\varLambda`| |$$\varXi$$ | `\varXi`| |$$\varPi$$ | `\varPi`| |$$\varSigma$$ | `\varSigma`| |$$\varUpsilon$$ | `\varUpsilon`| |$$\varPhi$$ | `\varPhi`| |$$\varPsi$$ | `\varPsi`| |$$\varOmega$$ | `\varOmega`| |$$\alpha$$ | `\alpha`| |$$\beta$$ | `\beta`| |$$\gamma$$ | `\gamma`| |$$\delta$$ | `\delta`| |$$\epsilon$$ | `\epsilon`| |$$\zeta$$ | `\zeta`| |$$\eta$$ | `\eta`| |$$\theta$$ | `\theta`| |$$\iota$$ | `\iota`| |$$\kappa$$ | `\kappa`| |$$\lambda$$ | `\lambda`| |$$\mu$$ | `\mu`| |$$\nu$$ | `\nu`| |$$\xi$$ | `\xi`| |$$\omicron$$ | `\omicron`| |$$\pi$$ | `\pi`| |$$\rho$$ | `\rho`| |$$\sigma$$ | `\sigma`| |$$\tau$$ | `\tau`| |$$\upsilon$$ | `\upsilon`| |$$\phi$$ | `\phi`| |$$\chi$$ | `\chi`| |$$\psi$$ | `\psi`| |$$\omega$$ | `\omega`| |$$\varepsilon$$ | `\varepsilon`| |$$\varkappa$$ | `\varkappa`| |$$\vartheta$$ | `\vartheta`| |$$\thetasym$$ | `\thetasym`| |$$\varpi$$ | `\varpi`| |$$\varrho$$ | `\varrho`| |$$\varsigma$$ | `\varsigma`| |$$\varphi$$ | `\varphi`| |$$\digamma$$ | `\digamma`|

$$\KaTeX$$ Greek Letters

|$$\KaTeX$$ |`Plain Text`| | ------- | ------- | |$$\imath$$ | `\imath`| |$$\nabla$$ | `\nabla`| |$$\Im$$ | `\Im`| |$$\Reals$$ | `\Reals`| |$$\text{\OE}$$ | `\text{\OE}`| |$$\jmath$$ | `\jmath`| |$$\partial$$ | `\partial`| |$$\image$$ | `\image`| |$$\wp$$ | `\wp`| |$$\text{\o}$$ | `\text{\o}`| |$$\aleph$$ | `\aleph`| |$$\Game$$ | `\Game`| |$$\Bbbk$$ | `\Bbbk`| |$$\weierp$$ | `\weierp`| |$$\text{\O}$$ | `\text{\O}`| |$$\alef$$ | `\alef`| |$$\Finv$$ | `\Finv`| |$$\N$$ | `\N`| |$$\Z$$ | `\Z`| |$$\text{\ss}$$ | `\text{\ss}`| |$$\alefsym$$ | `\alefsym`| |$$\cnums$$ | `\cnums`| |$$\natnums$$ | `\natnums`| |$$\text{\aa}$$ | `\text{\aa}`| |$$\text{\i}$$ | `\text{\i}`| |$$\beth$$ | `\beth`| |$$\Complex$$ | `\Complex`| |$$\R$$ | `\R`| |$$\text{\AA}$$ | `\text{\AA}`| |$$\text{\j}$$ | `\text{\j}`| |$$\gimel$$ | `\gimel`| |$$\ell$$ | `\ell`| |$$\Re$$ | `\Re`| |$$\text{\ae}$$ | `\text{\ae}`| |$$\daleth$$ | `\daleth`| |$$\hbar$$ | `\hbar`| |$$\real$$ | `\real`| |$$\text{\AE}$$ | `\text{\AE}`| |$$\eth$$ | `\eth`| |$$\hslash$$ | `\hslash`| |$$\reals$$ | `\reals`| |$$\text{\oe}$$ | `\text{\oe}`|

$$\KaTeX$$ Other Letters

University of Michigan - Ann Arbor

1. Operators
   * Big Operators
   * Binary Operators
   * Fractions and Binomials
   * Math Operators
2. Letters
   * Greek Letters
   * Other Letters
3. Logic and Set Theory Symbols
4. Accents and Symbol Decorators
5. Delimiters and Brackets
6. Matricies / Environments
7. Comparison and Relation Symbols
   * Equality Comparison Symbols
   * Negated Comparison / Relation Symbols
   * Arrows
8. Annotation Symbols
9. Misc Symbols and Punctuation
10. Overlap and Spacing Specifiers
11. Style, Font, Size and Color Modifiers

$$\KaTeX$$ Symbols and Syntax Reference Guide

1. Big Operators 
2. Binary Operators
3. Fractions and Binomials 
4. Math Operators 

Operators

1. Equality Comparison Symbols
2. Negated Relation Symbols 
3. Arrows 

Comparison and Relation Symbols

|$$\KaTeX$$ |`Plain Text`| | ------- | ------- | |$$\forall$$ | `\forall`| |$$\complement$$ | `\complement`| |$$\therefore$$ | `\therefore`| |$$\emptyset$$ | `\emptyset`| |$$\exists$$ | `\exists`| |$$\subset$$ | `\subset`| |$$\because$$ | `\because`| |$$\empty$$ | `\empty`| |$$\exist$$ | `\exist`| |$$\supset$$ | `\supset`| |$$\mapsto$$ | `\mapsto`| |$$\varnothing$$ | `\varnothing`| |$$\nexists$$ | `\nexists`| |$$\mid$$ | `\mid`| |$$\to$$ | `\to`| |$$\implies$$ | `\implies`| |$$\in$$ | `\in`| |$$\land$$ | `\land`| |$$\gets$$ | `\gets`| |$$\impliedby$$ | `\impliedby`| |$$\isin$$ | `\isin`| |$$\lor$$ | `\lor`| |$$\leftrightarrow$$ | `\leftrightarrow`| |$$\iff$$ | `\iff`| |$$\notin$$ | `\notin`| |$$\ni$$ | `\ni`| |$$\notni$$ | `\notni`| |$$\neg$$ | `\neg`| |$$\lnot$$ | `\lnot`|

Logic and Set Theory Symbols

|$$\KaTeX$$ |`Plain Text`| | ------- | ------- | |$$x_n$$ | `x_n`| |$$\stackrel{!}{=}$$ | `\stackrel{!}{=}`| |$$\overset{!}{=}$$ | `\overset{!}{=}`| |$$\underset{!}{=}$$ | `\underset{!}{=}`| |$${=}\mathllap{/\,}$$ | `{=}\mathllap{/\,}`| |$$\mathrlap{\,/}{=}$$ | `\mathrlap{\,/}{=}`| |$$a \atop b$$ | `a \atop b`| |$$a\raisebox{0.25em}{b}c$$ | `a\raisebox{0.25em}{b}c`| |$$_u^o$$ | `_u^o`| |$$e^x$$ | `e^x`| |$$\left(x^{\smash{2}}\right)$$ | `\left(x^{\smash{2}}\right)`| |$$\sqrt{\smash[b]{y}}$$ | `\sqrt{\smash[b]{y}}`| |$$\sum_{\mathclap{1\le i\le j\le n}} x_{ij}$$ | `\sum_{\mathclap{1\le i\le j\le n}} x_{ij}`|

$$\KaTeX$$ Overlap and Spacing Specifiers

| $$\KaTeX$$ | `Plain Text` |
| ------- | ------- |
| $$a'$$ | `a'` |
| $$\tilde{a}$$ | `\tilde{a}` |
| $$\mathring{g}$$ | `\mathring{g}` |
| $$a''$$ | `a''` |
| $$\widetilde{ac}$$ | `\widetilde{ac}` |
| $$\overgroup{AB}$$ | `\overgroup{AB}` |
| $$a^{\prime}$$ | `a^{\prime}` |
| $$\utilde{AB}$$ | `\utilde{AB}` |
| $$\undergroup{AB}$$ | `\undergroup{AB}` |
| $$\acute{a}$$ | `\acute{a}` |
| $$\vec{F}$$ | `\vec{F}` |
| $$\Overrightarrow{AB}$$ | `\Overrightarrow{AB}` |
| $$\bar{y}$$ | `\bar{y}` |
| $$\overleftarrow{AB}$$ | `\overleftarrow{AB}` |
| $$\overrightarrow{AB}$$ | `\overrightarrow{AB}` |
| $$\breve{a}$$ | `\breve{a}` |
| $$\underleftarrow{AB}$$ | `\underleftarrow{AB}` |
| $$\underrightarrow{AB}$$ | `\underrightarrow{AB}` |
| $$\check{a}$$ | `\check{a}` |
| $$\overleftharpoon{ac}$$ | `\overleftharpoon{ac}` |
| $$\overrightharpoon{ac}$$ | `\overrightharpoon{ac}` |
| $$\dot{a}$$ | `\dot{a}` |
| $$\overleftrightarrow{AB}$$ | `\overleftrightarrow{AB}` |
| $$\overbrace{AB}$$ | `\overbrace{AB}` |
| $$\ddot{a}$$ | `\ddot{a}` |
| $$\underleftrightarrow{AB}$$ | `\underleftrightarrow{AB}` |
| $$\underbrace{AB}$$ | `\underbrace{AB}` |
| $$\grave{a}$$ | `\grave{a}` |
| $$\overline{AB}$$ | `\overline{AB}` |
| $$\overlinesegment{AB}$$ | `\overlinesegment{AB}` |
| $$\hat{\theta}$$ | `\hat{\theta}` |
| $$\underline{AB}$$ | `\underline{AB}` |
| $$\underlinesegment{AB}$$ | `\underlinesegment{AB}` |
| $$\widehat{ac}$$ | `\widehat{ac}` |
| $$\widecheck{ac}$$ | `\widecheck{ac}` |

$$\KaTeX$$ Accents and Symbol Decorators

|$$\KaTeX$$ |`Plain Text`|
| ------- | ------- |
|$$\color{blue} F=ma$$ | `\color{blue} F=ma`|
|$$\textcolor{blue}{F=ma}$$ | `\textcolor{blue}{F=ma}`|
|$$\textcolor{#228B22}{F=ma}$$ | `\textcolor{#228B22}{F=ma}`|
|$$\colorbox{aqua}{A}$$ | `\colorbox{aqua}{A}`|
|$$\fcolorbox{red}{aqua}{A}$$ | `\fcolorbox{red}{aqua}{A}`|
|$$\mathrm{Ab0}$$ | `\mathrm{Ab0}`|
|$$\mathbf{Ab0}$$ | `\mathbf{Ab0}`|
|$$\mathit{Ab0}$$ | `\mathit{Ab0}`|
|$$\mathnormal{Ab0}$$ | `\mathnormal{Ab0}`|
|$$\textbf{Ab0}$$ | `\textbf{Ab0}`|
|$$\textit{Ab0}$$ | `\textit{Ab0}`|
|$$\textrm{Ab0}$$ | `\textrm{Ab0}`|
|$$\bf Ab0$$ | `\bf Ab0`|
|$$\it Ab0$$ | `\it Ab0`|
|$$\rm Ab0$$ | `\rm Ab0`|
|$$\bold{Ab0}$$ | `\bold{Ab0}`|
|$$\textup{Ab0}$$ | `\textup{Ab0}`|
|$$\textnormal{Ab0}$$ | `\textnormal{Ab0}`|
|$$\boldsymbol{Ab}$$ | `\boldsymbol{Ab}`|
|$$\Bbb{AB}$$ | `\Bbb{AB}`|
|$$\text{Ab0}$$ | `\text{Ab0}`|
|$$\bm{Ab0}$$ | `\bm{Ab0}`|
|$$\mathbb{AB}$$ | `\mathbb{AB}`|
|$$\mathsf{Ab0}$$ | `\mathsf{Ab0}`|
|$$\textmd{Ab0}$$ | `\textmd{Ab0}`|
|$$\frak{Ab0}$$ | `\frak{Ab0}`|
|$$\textsf{Ab0}$$ | `\textsf{Ab0}`|
|$$\mathtt{Ab0}$$ | `\mathtt{Ab0}`|
|$$\mathfrak{Ab0}$$ | `\mathfrak{Ab0}`|
|$$\sf Ab0$$ | `\sf Ab0`|
|$$\texttt{Ab0}$$ | `\texttt{Ab0}`|
|$$\mathcal{AB0}$$ | `\mathcal{AB0}`|
|$$\tt Ab0$$ | `\tt Ab0`|
|$$\mathscr{AB}$$ | `\mathscr{AB}`|
|$$\textsf{\textbf{H}}$$ | `\textsf{\textbf{H}}`|
|$$\mathsf{\mathbf{H}}$$ | `\mathsf{\mathbf{H}}`|
|$$\pmb{\mu}$$ | `\pmb{\mu}`|
|$$\Huge AB$$ | `\Huge AB`|
|$$\normalsize AB$$ | `\normalsize AB`|
|$$\huge AB$$ | `\huge AB`|
|$$\small AB$$ | `\small AB`|
|$$\LARGE AB$$ | `\LARGE AB`|
|$$\footnotesize AB$$ | `\footnotesize AB`|
|$$\Large AB$$ | `\Large AB`|
|$$\scriptsize AB$$ | `\scriptsize AB`|
|$$\large AB$$ | `\large AB`|
|$$\tiny AB$$ | `\tiny AB`|
|$$\displaystyle\sum_{i=1}^n$$ | `\displaystyle\sum_{i=1}^n`|
|$$\textstyle\sum_{i=1}^n$$ | `\textstyle\sum_{i=1}^n`|
|$$\scriptstyle x$$ | `\scriptstyle x`|
|$$\scriptscriptstyle x$$ | `\scriptscriptstyle x`|
|$$\lim\limits_x$$ | `\lim\limits_x`|
|$$\lim\nolimits_x$$ | `\lim\nolimits_x`|
|$$\verb!x^2!$$ | `\verb!x^2!`|

$$\KaTeX$$ Style, Font, Size and Color Modifiers

|$$\KaTeX$$ |`Plain Text`| | ------- | ------- | |$$\cancel{5}$$ | `\cancel{5}`| |$$\bcancel{5}$$ | `\bcancel{5}`| |$$\overbrace{a+b+c}^{\text{note}}$$ | `\overbrace{a+b+c}^{\text{note}}`| |$$\underbrace{a+b+c}_{\text{note}}$$ | `\underbrace{a+b+c}_{\text{note}}`| |$$\xcancel{ABC}$$ | `\xcancel{ABC}`| |$$\sout{abc}$$ | `\sout{abc}`| |$$\not =$$ | `\not =`| |$$\boxed{\pi=\frac c d}$$ | `\boxed{\pi=\frac c d}`|

$$\KaTeX$$ Annotation Symbols

|$$\KaTeX$$ |`Plain Text`| | ------- | ------- | |$$( )$$ | `( )`| |$$\lparen$$ | `\lparen`| |$$\rparen$$ | `\rparen`| |$$⌈ ⌉$$ | `⌈ ⌉`| |$$\lceil$$ | `\lceil`| |$$\rceil$$ | `\rceil`| |$$\uparrow$$ | `\uparrow`| |$$[ ]$$ | `[ ]`| |$$\lbrack$$ | `\lbrack`| |$$\rbrack$$ | `\rbrack`| |$$⌊ ⌋$$ | `⌊ ⌋`| |$$\lfloor$$ | `\lfloor`| |$$\rfloor$$ | `\rfloor`| |$$\downarrow$$ | `\downarrow`| |$$\{ \}$$ | `\{ \}`| |$$\lbrace$$ | `\lbrace`| |$$\rbrace$$ | `\rbrace`| |$$⎰⎱$$ | `⎰⎱`| |$$\lmoustache$$ | `\lmoustache`| |$$\rmoustache$$ | `\rmoustache`| |$$\updownarrow$$ | `\updownarrow`| |$$⟨ ⟩$$ | `⟨ ⟩`| |$$\langle$$ | `\langle`| |$$\rangle$$ | `\rangle`| |$$⟮ ⟯$$ | `⟮ ⟯`| |$$\lgroup$$ | `\lgroup`| |$$\rgroup$$ | `\rgroup`| |$$\Uparrow$$ | `\Uparrow`| |$$|$$ | `|`| |$$\vert$$ | `\vert`| |$$┌ ┐$$ | `┌ ┐`| |$$\ulcorner$$ | `\ulcorner`| |$$\urcorner$$ | `\urcorner`| |$$\Downarrow$$ | `\Downarrow`| |$$\|$$ | `\|`| |$$\Vert$$ | `\Vert`| |$$└ ┘$$ | `└ ┘`| |$$\llcorner$$ | `\llcorner`| |$$\lrcorner$$ | `\lrcorner`| |$$\Updownarrow$$ | `\Updownarrow`| |$$\lvert$$ | `\lvert`| |$$\rvert$$ | `\rvert`| |$$\lVert$$ | `\lVert`| |$$\rVert$$ | `\rVert`| |$$\backslash$$ | `\backslash`| |$$\lang$$ | `\lang`| |$$\rang$$ | `\rang`| |$$\lt \gt$$ | `\lt \gt`| |$$⟦ ⟧$$ | `⟦ ⟧`| |$$\llbracket$$ | `\llbracket`| |$$\rrbracket$$ | `\rrbracket`| |$$\lBrace \rBrace$$ | `\lBrace \rBrace`|

$$\KaTeX$$ Brackets and Delimiters

$$\KaTeX$$ Misc Symbols and Punctuation

| $$\KaTeX$$ | `Plain Text` |
| ------- | ------- |
| $$\begin{matrix}a & b \\c & d\end{matrix}$$ | `\begin{matrix}a & b \\c & d\end{matrix}` |
| $$\begin{array}{cc}a & b \\c & d\end{array}$$ | `\begin{array}{cc}a & b \\c & d\end{array}` |
| $$\begin{pmatrix}a & b \\c & d\end{pmatrix}$$ | `\begin{pmatrix}a & b \\c & d\end{pmatrix}` |
| $$\begin{bmatrix}a & b \\c & d\end{bmatrix}$$ | `\begin{bmatrix}a & b \\c & d\end{bmatrix}` |
| $$\begin{vmatrix}a & b \\c & d\end{vmatrix}$$ | `\begin{vmatrix}a & b \\c & d\end{vmatrix}` |
| $$\begin{Vmatrix}a & b \\c & d\end{Vmatrix}$$ | `\begin{Vmatrix}a & b \\c & d\end{Vmatrix}` |
| $$\begin{Bmatrix}a & b \\c & d\end{Bmatrix}$$ | `\begin{Bmatrix}a & b \\c & d\end{Bmatrix}` |
| $$\begin{array}{c:c:c}a & b & c \\ \hline d & e & f \\ \hdashline g & h & i\end{array}$$ | `\begin{array}{c:c:c}a & b & c \\ \hline d & e & f \\ \hdashline g & h & i\end{array}` |
| $$\begin{aligned}a&=b+c \\d+e&=f\end{aligned}$$ | `\begin{aligned}a&=b+c \\d+e&=f\end{aligned}` |
| $$\begin{alignedat}{2} 10&x+ &3&y = 2 \\3&x+&13&y = 4\end{alignedat}$$ | `\begin{alignedat}{2} 10&x+ &3&y = 2 \\3&x+&13&y = 4\end{alignedat}` |
| $$\begin{gathered}a=b \\e=b+c\end{gathered}$$ | `\begin{gathered}a=b \\e=b+c\end{gathered}` |
| $$x = \begin{cases}a &\text{if } b \\c &\text{if } d\end{cases}$$ | `x = \begin{cases}a &\text{if } b \\c &\text{if } d\end{cases}` |

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